Numeration systems where the basis is defined by a linear recurrence with integer coefficients are considered. A rewriting system is associated with the recurrence. In this paper we study the case when it is confluent. We prove that the function of normalization which transforms any representation of an integer into the normal one — obtained by the usual algorithm — can be realized by a finite automation which is the composition of a left subsequential transducer and of a right subsequential transducer associated with the rewriting system. Addition of integers in a confluent linear numeration system is also computable by a finite automaton. These results extend to the representation of real numbers in basis θ, where θ is the dominant root of the characteristic polynomial of the recurrence.